Suboptimal Polynomial Meshes on Planar Lipschitz Domains

We construct norming meshes with cardinality 𝒪(n ^s ), s = 3, for polynomials of total degree at most n on the closure of bounded planar Lipschitz domains. Such cardinality is intermediate between optimality (s = 2), recently obtained by Kroó on multidimensional C ² star-like domains, and that arising from a general construction on Markov compact sets due to Calvi and Levenberg (s = 4).     

Standard Classes on the Blow-Up of ℙ n at Points in Very General Position

We study (?1)-classes on the blow-up X of ? ⁿ at points in very general position; in the case n = 2, we give a new proof of the equivalence of two conjectures about the dimension of the class of a divisor on X, and we prove that h ¹(D) = 0 for any nef D which is a nonnegative sum of (?1)-curves. For n = 3, we fill a gap in the formulation of a conjecture about the dimension of a class. We provide algorithms and Maple code based on these conjectures.     

Minimal graded free resolutions for monomial curves in 𝔸4 defined by almost arithmetic sequences

Let m = (m₀, m₁, m₂, n) be an almost arithmetic sequence, i.e., a sequence of positive integers with gcd(m₀, m₁, m₂, n) = 1, such that m₀ < m₁ < m₂ form an arithmetic progression, n is arbitrary and they minimally generate the numerical semigroup Γ =m₀? +m₁? +m₂? +n?. Let k be a field. The homogeneous coordinate ring k[Γ] of the affine monomial curve parametrically defined by X₀ = t^m₀, X₁ = t^m₁, X₂ = t^m₂, Y = tⁿ is a graded R-module, where R is the polynomial ring k[X₀, X₁, X₂, Y] with the grading degX_i: = m_i, degY: = n. In this paper, we construct a minimal graded free resolution for k[Γ].     

Hilbert Series and Lefschetz Properties of Dimension One Almost Complete Intersections

We generalize some results about the graded Milnor algebras of projective hypersurfaces with isolated singularities to the more general case of an almost complete intersection ideal J of dimension one. When the saturation I of J is a complete intersection, we get formulas for some invariants. Examples of hypersurfaces V: f = 0 in ?ⁿ whose Jacobian ideals J satisfy this property and with nontrivial Alexander polynomials are given in any dimension. A Lefschetz property for the graded quotient I/J is proved for n = 2 and a counterexample due to A. Conca is given for such a property when n = 3.     

Partial Regularity for Stationary Solutions to Liouville-Type Equation in Dimension 3

In dimension n = 3, we prove that the singular set of any stationary solution to the Liouville equation ? Δ u = e ^u , which belongs to W ^1,2, has Hausdorff dimension at most 1.     

Presimplifiable and Cyclic Group Rings

Let R be any commutative ring with identity, and let C be a (finite or infinite) cyclic group. We show that the group ring R(C) is presimplifiable if and only if its augmentation ideal I(C) is presimplifiable. We conjecture that the group rings R(C _n ) are presimplifiable if and only if n = p ^m , p ∈ J(R), p is prime, and R is presimplifiable. We show the necessity of n = p ^m , and we prove the sufficiency when n = 2, 3, 4. These results were made possible by a new formula derived herein for the circulant determinantal coefficients.     

Projective Varieties Swept Out by Rational Normal Curves

Let X ? ? ⁿ be a complex nondegenerate projective variety of dimension m ≥ 2. For t ≤ n ? m and a general q ∈ ? ⁿ , the linear space L _q spanned by q and t general points of X meets X in a finite set of points. We classify those X ? ? ⁿ for which there exists a point q ∈ ? ⁿ such that L _q meets X in a positive dimensional variety. If this occurs, there exists d ≤ n ? m such that a degree d rational normal curve through d general points of X is contained in X. Examples of this situation are provided. An infinitesimal generalization of part of the main result is also stated.     

Zero-Semidistributive Inverse Semigroups

An inverse semigroup S is said to be 0-semidistributive if its lattice ?F (S) of full inverse subsemigroups is 0-semidistributive. We show that it is sufficient to study simple inverse semigroups which are not groups. Our main theorem states that such a simple inverse semigroup S is 0-semidistributive if and only if (1) S is E-unitary, (2) S is aperiodic, (3) for any a,b ∈ S/σ with ab ≠ 1, there exist nonzero integers n and m such that (ab) ^m  = a ⁿ or (ab) ^m  = b ⁿ , where σ is the minimum group congruence on S.     

A Note on Projective and Flat Dimensions and the Bieri-Neumann-Strebel-Renz Σ-Invariants

Let G be a finitely generated group, and A a ?[G]-module of flat dimension n such that the homological invariant Σ ⁿ (G, A) is not empty. We show that A has projective dimension n as a ?[G]-module. In particular, if G is a group of homological dimension hd(G) = n such that the homological invariant Σ ⁿ (G, ?) is not empty, then G has cohomological dimension cd(G) = n. We show that if G is a finitely generated soluble group, the converse is true subject to taking a subgroup of finite index, i.e., the equality cd (G) = hd(G) implies that there is a subgroup H of finite index in G such that Σ^∞(H, ?) ≠ ?.     

Linear preservers of matrices of rank-2

Let T be a linear operator on the space of all m×n matrices over any field. we prove that if T maps rank-2 matrices to rank-2 matrices then there exist nonsingular matrices U and V such that either T(X)=UXV for all matrices X, or m=n and T(X)=UX^tV for all matrices X where X^t denotes the transpose of X.     

Real Hypersurfaces Having Many Pseudo-hyperplanes

Abstract

Let nand dbe natural integers satisfying n ≥ 3 and d ≥ 10. Let Xbe an irreducible real hypersurface Xin ? ⁿ of degree dhaving many pseudo-hyperplanes. Suppose that Xis not a projective cone. We show that the arrangement ? of all d ? 2 pseudo-hyperplanes of Xis trivial, i.e., there is a real projective linear subspace Lof ? ⁿ (?) of dimension n ? 2 such that L ? Hfor all H ∈ ?. As a consequence, the normalization of Xis fibered over ?¹in quadrics. Both statements are in sharp contrast with the case n = 2; the first statement also shows that there is no Brusotti-type result for hypersurfaces in ? ⁿ , for n ≥ 3.     

On the Semigroup Algebra of Binary Relations

The semigroup of binary relations on {1,…, n} with the relative product is isomorphic to the semigroup B _n of n × n zero-one matrices with the Boolean matrix product. Over any field F, we prove that the semigroup algebra FB _n contains an ideal K _n of dimension (2 ⁿ  ? 1)², and we construct an explicit isomorphism of K _n with the matrix algebra M _{2 ⁿ ?1}(F).     

The DJL Conjecture for CP Matrices over Special Inclines

Drew, Johnson, and Loewy conjectured that for n ≥ 4, the CP-rank of every n × n completely positive real matrix is at most [n²/4]. While this conjecture has recently been disproved for completely positive real matrices, we show that this conjecture is true for n × n completely positive matrices over certain special types of inclines. In addition, we prove an incline version of Markham's theorems which gives sufficient conditions for completely positive matrices over special inclines to have triangular factorizations.     

Osculating Degeneration of Curves

Abstract

The main objects of this paper are osculating spaces of order mto smooth algebraic curves, with the property of meeting the curve again. We prove that the only irreducible curves with an infinite number of this type of osculating spaces of order mare curves in P ^m+1whose degree nis greater than m + 1. This is a generalization of the result and proof of Kaji (Kaji, H. (1986). On the tangentially degenerate curves. J. London Math. Soc.33(2):430–440) that corresponds to the case m = 1. We also obtain an enumerative formula for the number of those osculating spaces to curves in P ^m+2. The case m = 1 of it is a classical formula proved with modern techniques by Le Barz (Le Barz, P. (1982). Formules multisécantes pour les courbes gauches quelconques. In: Enumerative Geometry and Classical Algebraic Geometry. Prog. in Mathematics 24, Birkhäuser, pp. 165–197).     

Low Frequency Estimates and Local Energy Decay for Asymptotically Euclidean Laplacians

For Riemannian metrics G on ? ^d which are long range perturbations of the flat one, we prove estimates for (? Δ _G  ? λ ?iε)^?n as λ → 0, which are uniform with respect to ε, for all n ≤ [d/2] +1 in odd dimension and n ≤ d/2 in even dimension. We also give applications to the time decay of Schrödinger and Wave (or Klein–Gordon) equations.